We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic d-space and Coxeter's regular hyperbolic honeycombs (with infinite or star-shaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice. In section 2 the last remaining 2-dimensional case is decided: for any odd m>6, star-honeycombs {m, m/2} are embeddable while {m/2, m} are not (unique case of non-embedding for dimension 2). As a spherical analogue of those honeycombs, we enumerate, in section 3, 36 Riemann surfaces representing all nine regular polyhedra on the sphere. In section 4, non-embeddability of all remaining star-honeycombs (on 3-sphere and hyperbolic 4-space) is proved. In the last section 5, all cases of embedding for dimension d>2 are identified. Besides hyper-simplices and hyper-octahedra, they are exactly those with bipartite skeleton: hyper-cubes, cubic lattices and 8, 2, 1 tilings of hyperbolic 3-, 4-, 5-space (only two, {435} and {4335}, of those 11 are compact).
{"title":"Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices","authors":"M. Deza, M. Shtogrin","doi":"10.2969/ASPM/02710073","DOIUrl":"https://doi.org/10.2969/ASPM/02710073","url":null,"abstract":"We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic d-space and Coxeter's regular hyperbolic honeycombs (with infinite or star-shaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice. In section 2 the last remaining 2-dimensional case is decided: for any odd m>6, star-honeycombs {m, m/2} are embeddable while {m/2, m} are not (unique case of non-embedding for dimension 2). As a spherical analogue of those honeycombs, we enumerate, in section 3, 36 Riemann surfaces representing all nine regular polyhedra on the sphere. In section 4, non-embeddability of all remaining star-honeycombs (on 3-sphere and hyperbolic 4-space) is proved. In the last section 5, all cases of embedding for dimension d>2 are identified. Besides hyper-simplices and hyper-octahedra, they are exactly those with bipartite skeleton: hyper-cubes, cubic lattices and 8, 2, 1 tilings of hyperbolic 3-, 4-, 5-space (only two, {435} and {4335}, of those 11 are compact).","PeriodicalId":192449,"journal":{"name":"Arrangements–Tokyo 1998","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116559387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We relate the cohomology of the Orlik-Solomon algebra of a discriminantal arrangement to the local system cohomology of the complement. The Orlik-Solomon algebra of such an arrangement (viewed as a complex) is shown to be a linear approximation of a complex arising from the fundamental group of the complement, the cohomology of which is isomorphic to that of the complement with coefficients in an arbitrary complex rank one local system. We also establish the relationship between the cohomology support loci of the complement of a discriminantal arrangement and the resonant varieties of its Orlik-Solomon algebra. Department of Mathematics Louisiana State University Baton Rouge, LA 70803 U. S. A. cohen@math.lsu.edu http://www.math.lsu.edu/~cohen 1991 Mathematics Subject Classification. Primary 52B30, 55N25; Secondary 20F36.
{"title":"On the Cohomology of Discriminantal Arrangements and Orlik–Solomon Algebras","authors":"Daniel C. Cohen","doi":"10.2969/ASPM/02710027","DOIUrl":"https://doi.org/10.2969/ASPM/02710027","url":null,"abstract":"We relate the cohomology of the Orlik-Solomon algebra of a discriminantal arrangement to the local system cohomology of the complement. The Orlik-Solomon algebra of such an arrangement (viewed as a complex) is shown to be a linear approximation of a complex arising from the fundamental group of the complement, the cohomology of which is isomorphic to that of the complement with coefficients in an arbitrary complex rank one local system. We also establish the relationship between the cohomology support loci of the complement of a discriminantal arrangement and the resonant varieties of its Orlik-Solomon algebra. Department of Mathematics Louisiana State University Baton Rouge, LA 70803 U. S. A. cohen@math.lsu.edu http://www.math.lsu.edu/~cohen 1991 Mathematics Subject Classification. Primary 52B30, 55N25; Secondary 20F36.","PeriodicalId":192449,"journal":{"name":"Arrangements–Tokyo 1998","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1999-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127431393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H �2 (X), to the second nilpotent quotient, G/G3. We define invariants of G/G3 by counting normal subgroups of a fixed prime index p, according to their abelianization. We show how to compute this distribution from the resonance varieties of the Orlik-Solomon algebra mod p. As an application, we establish the cohomology classification of 2-arrangements of n� 6 planes in R 4 .
{"title":"Cohomology rings and nilpotent quotients of real and complex arrangements","authors":"D. Matei, Alexander I. Suciu","doi":"10.2969/ASPM/02710185","DOIUrl":"https://doi.org/10.2969/ASPM/02710185","url":null,"abstract":"For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H �2 (X), to the second nilpotent quotient, G/G3. We define invariants of G/G3 by counting normal subgroups of a fixed prime index p, according to their abelianization. We show how to compute this distribution from the resonance varieties of the Orlik-Solomon algebra mod p. As an application, we establish the cohomology classification of 2-arrangements of n� 6 planes in R 4 .","PeriodicalId":192449,"journal":{"name":"Arrangements–Tokyo 1998","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1998-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123378336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. The classical harmonic functions are characterized in terms of the mean value property with respect to the unit ball. Replacing the ball by a polytope, we are led to the notion of polyhedral har monic functions, i.e., those continuous functions which satisfy the mean value property with respect to a given polytope. The study of polyhedral harmonic functions involves not only analysis but also algebra, including combinatorics of polytopes and invariant theory for finite reflection groups. We give a brief survey on this subject, focusing on some recent results obtained by the author.
{"title":"Polytopes, Invariants and Harmonic Functions","authors":"Katsunori Iwasaki","doi":"10.2969/ASPM/02710145","DOIUrl":"https://doi.org/10.2969/ASPM/02710145","url":null,"abstract":". The classical harmonic functions are characterized in terms of the mean value property with respect to the unit ball. Replacing the ball by a polytope, we are led to the notion of polyhedral har monic functions, i.e., those continuous functions which satisfy the mean value property with respect to a given polytope. The study of polyhedral harmonic functions involves not only analysis but also algebra, including combinatorics of polytopes and invariant theory for finite reflection groups. We give a brief survey on this subject, focusing on some recent results obtained by the author.","PeriodicalId":192449,"journal":{"name":"Arrangements–Tokyo 1998","volume":"469 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116080641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a real hyperplane arrangement A ⊂ R, among the first invariants that were determined for A were the number of chambers in the complement RnA by Zavslavsky [Za] and the number of bounded chambers by Crapo [Cr]. In the consideration of certain classes of hypergeometric functions, there also arise arrangements of hypersurfaces which need not be hyperplanes (see e.g. Aomoto [Ao]). In this paper we will obtain a formula for the number of bounded regions (i.e. chambers) in the complement of a nonlinear arrangement of hypersurfaces. For example, for the general position arrangements of quadrics in Figure 1, we see the number of bounded regions in the complement are respectively 1, 5, and 13.
{"title":"On the number of Bounding Cycles for Nonlinear Arrangements","authors":"J. Damon","doi":"10.2969/ASPM/02710051","DOIUrl":"https://doi.org/10.2969/ASPM/02710051","url":null,"abstract":"For a real hyperplane arrangement A ⊂ R, among the first invariants that were determined for A were the number of chambers in the complement RnA by Zavslavsky [Za] and the number of bounded chambers by Crapo [Cr]. In the consideration of certain classes of hypergeometric functions, there also arise arrangements of hypersurfaces which need not be hyperplanes (see e.g. Aomoto [Ao]). In this paper we will obtain a formula for the number of bounded regions (i.e. chambers) in the complement of a nonlinear arrangement of hypersurfaces. For example, for the general position arrangements of quadrics in Figure 1, we see the number of bounded regions in the complement are respectively 1, 5, and 13.","PeriodicalId":192449,"journal":{"name":"Arrangements–Tokyo 1998","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114079614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Consider the set of surface-curve pairs (X,C), where X is a normal surface and C is an algebraic curve. In this paper, we de fine a family :F of normal surface-curve pairs, which is closed under coverings, and which contains all smooth surface-curve pairs (X, C), where X is smooth and C has smooth irreducible components with normal crossings. We give a modification of W. Neumann's defini tion of plumbing graphs, their associated 3-dimensional graph mani folds, and intersection matrices, and use this construction to describe rational intersection matrices and boundary manifolds for regular branched coverings.
{"title":"Plumbing Graphs for Normal Surface-Curve Pairs","authors":"E. Hironaka","doi":"10.2969/ASPM/02710127","DOIUrl":"https://doi.org/10.2969/ASPM/02710127","url":null,"abstract":". Consider the set of surface-curve pairs (X,C), where X is a normal surface and C is an algebraic curve. In this paper, we de fine a family :F of normal surface-curve pairs, which is closed under coverings, and which contains all smooth surface-curve pairs (X, C), where X is smooth and C has smooth irreducible components with normal crossings. We give a modification of W. Neumann's defini tion of plumbing graphs, their associated 3-dimensional graph mani folds, and intersection matrices, and use this construction to describe rational intersection matrices and boundary manifolds for regular branched coverings.","PeriodicalId":192449,"journal":{"name":"Arrangements–Tokyo 1998","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121481003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This survey is intended to provide a background for the authors paper [23]. The latter was the subject of the talk given by the second author at the Arrangement Workshop. The central theme of this survey is the cohomology of local systems on quasi-projective varieties, especially on the complements to algebraic curves and arrangements of lines in P 2 . A few of the results of [23] are discussed in section 4 while the first part of this paper contains some of highlights of Deligne's theory [7] and several examples from the theory of Alexander invariants developed mostly by the first author in the series of papers [17] [22]. We also included several problems indicating possible further development. The second author uses the opportunity to thank M. Oka and H. Terao for the hard labor of organizing the Arrangement Workshop.
{"title":"Cohomology of Local systems","authors":"A. Libgober, S. Yuzvinsky","doi":"10.2969/ASPM/02710169","DOIUrl":"https://doi.org/10.2969/ASPM/02710169","url":null,"abstract":"This survey is intended to provide a background for the authors paper [23]. The latter was the subject of the talk given by the second author at the Arrangement Workshop. The central theme of this survey is the cohomology of local systems on quasi-projective varieties, especially on the complements to algebraic curves and arrangements of lines in P 2 . A few of the results of [23] are discussed in section 4 while the first part of this paper contains some of highlights of Deligne's theory [7] and several examples from the theory of Alexander invariants developed mostly by the first author in the series of papers [17] [22]. We also included several problems indicating possible further development. The second author uses the opportunity to thank M. Oka and H. Terao for the hard labor of organizing the Arrangement Workshop.","PeriodicalId":192449,"journal":{"name":"Arrangements–Tokyo 1998","volume":"80 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129610206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
If K is C, then the complement M(A) is an open and connected subset ofV. The present paper is concerned with fundamental groups of complements of complex arrangements of hyperplanes. The most popular such a group is certainly the pure braid group; it appears as the fundamental group of the complement of the "braid arrangement" (see [OT]). So, n1(M(A)) can be considered as a generalization of the pure braid group, and one can expect to show that many properties of the pure braid group also hold for n1 ( M (A)). However, the only general known results on this group are presentations [Ar], [CSl], [Ra], [Sal]. Many interesting questions remain, for example, to know whether such a group is torsion free. We focus in this paper on two families of arrangements of hyperplanes, to the fundamental group of which many well-known results on the pure braid group can be extended. Both of them, of course, contain the braid arrangement. These families are the "simplicial arrangements" and the "supersolvable arrangements". Note that there is another wellunderstood family of arrangements, the "reflection arrangements" (see
{"title":"On the fundamental group of the complement of a complex hyperplane arrangement","authors":"L. Paris","doi":"10.2969/ASPM/02710257","DOIUrl":"https://doi.org/10.2969/ASPM/02710257","url":null,"abstract":"If K is C, then the complement M(A) is an open and connected subset ofV. The present paper is concerned with fundamental groups of complements of complex arrangements of hyperplanes. The most popular such a group is certainly the pure braid group; it appears as the fundamental group of the complement of the \"braid arrangement\" (see [OT]). So, n1(M(A)) can be considered as a generalization of the pure braid group, and one can expect to show that many properties of the pure braid group also hold for n1 ( M (A)). However, the only general known results on this group are presentations [Ar], [CSl], [Ra], [Sal]. Many interesting questions remain, for example, to know whether such a group is torsion free. We focus in this paper on two families of arrangements of hyperplanes, to the fundamental group of which many well-known results on the pure braid group can be extended. Both of them, of course, contain the braid arrangement. These families are the \"simplicial arrangements\" and the \"supersolvable arrangements\". Note that there is another wellunderstood family of arrangements, the \"reflection arrangements\" (see","PeriodicalId":192449,"journal":{"name":"Arrangements–Tokyo 1998","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114575495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The notion of finite type invariants of knots was introduced by Vassiliev in his study of the discriminats of function spaces (see [13]). It was shown by Kontsevich [9] that such invariants, which we shall call the Vassiliev invariants, can be expressed universally by iterated integrals of logarithmic forms on the configuration space of distinct points in the complex plane. In the present paper we focus on the Vassiliev invariants of braids. Our main object is to clarify the relation between the Vassiliev invariants of braids and the iterated integrals of logarithmic forms on the configuration space which are homotopy invariant. A version of such description for pure braids is given in [6]. We denote by Bn the braid group on n strings. Let J be the ideal of the group ring C[Bn] generated by ai a;1, where { ai}i::;i::;n-1 is the set of standard generators of Bn. The vector space of the Vassiliev invariants of Bn of order k with values in C can be identified with Hom(C[Bn]/ Jk+l, C). Let us stress that such vector space had been studied in terms of the iterated integrals due to K. T. Chen before the work of Vassiliev. We introduce a graded algebra An, which is a semi-direct product of the completed universal enveloping algebra of the holonomy Lie algebra of the configuration space and the group algebra of the symmetric group. We construct a homomorphism 0 : Bn --+ An expressed as an infinite sum of Chen's iterated integrals, which gives a universal expression of the holonomy of logarithmic connections. This homomorphism may be considered as a prototype of the Kontsevich integral for knots. Using this homomorpshim we shall determine all iterated integrals of logarithmic forms which provide invariants of braids (see Theorem 3.1). As a Corollary we recover the isomorphism
{"title":"Vassiliev Invariants of Braids and Iterated Integrals","authors":"T. Kohno","doi":"10.2969/ASPM/02710157","DOIUrl":"https://doi.org/10.2969/ASPM/02710157","url":null,"abstract":"The notion of finite type invariants of knots was introduced by Vassiliev in his study of the discriminats of function spaces (see [13]). It was shown by Kontsevich [9] that such invariants, which we shall call the Vassiliev invariants, can be expressed universally by iterated integrals of logarithmic forms on the configuration space of distinct points in the complex plane. In the present paper we focus on the Vassiliev invariants of braids. Our main object is to clarify the relation between the Vassiliev invariants of braids and the iterated integrals of logarithmic forms on the configuration space which are homotopy invariant. A version of such description for pure braids is given in [6]. We denote by Bn the braid group on n strings. Let J be the ideal of the group ring C[Bn] generated by ai a;1, where { ai}i::;i::;n-1 is the set of standard generators of Bn. The vector space of the Vassiliev invariants of Bn of order k with values in C can be identified with Hom(C[Bn]/ Jk+l, C). Let us stress that such vector space had been studied in terms of the iterated integrals due to K. T. Chen before the work of Vassiliev. We introduce a graded algebra An, which is a semi-direct product of the completed universal enveloping algebra of the holonomy Lie algebra of the configuration space and the group algebra of the symmetric group. We construct a homomorphism 0 : Bn --+ An expressed as an infinite sum of Chen's iterated integrals, which gives a universal expression of the holonomy of logarithmic connections. This homomorphism may be considered as a prototype of the Kontsevich integral for knots. Using this homomorpshim we shall determine all iterated integrals of logarithmic forms which provide invariants of braids (see Theorem 3.1). As a Corollary we recover the isomorphism","PeriodicalId":192449,"journal":{"name":"Arrangements–Tokyo 1998","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129276707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
are the Gamma and the Beta functions. In this paper, we give a geometric meaning for these formulae: If one regards such an integral as the dual pairing between a (kind of) cycle and a (kind of) differential form, then the value given in the right hand side of each formula is the product of the intersection numbers of the two cycles and that of the two forms appeared in the left-hand side. Of course the intersection theory is not made only to explain these well known formulae; for applications, see [CM], [KM], [Yl].
{"title":"Recent progress of intersection theory for twisted (co)homology groups","authors":"Keiji Matsumoto, Masaaki Yoshida","doi":"10.2969/ASPM/02710217","DOIUrl":"https://doi.org/10.2969/ASPM/02710217","url":null,"abstract":"are the Gamma and the Beta functions. In this paper, we give a geometric meaning for these formulae: If one regards such an integral as the dual pairing between a (kind of) cycle and a (kind of) differential form, then the value given in the right hand side of each formula is the product of the intersection numbers of the two cycles and that of the two forms appeared in the left-hand side. Of course the intersection theory is not made only to explain these well known formulae; for applications, see [CM], [KM], [Yl].","PeriodicalId":192449,"journal":{"name":"Arrangements–Tokyo 1998","volume":"146 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133692730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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